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van der Waals^{a}  R_{v}  D_{v}  Bond^{b}  r_{e}  k_{r}  E_{bond} 
C  3.8050  0.06920  CC  1.4114  720.000  133.0000 
Angle^{c}  k_{r1r2}  
CCC  120.000  196.130  72.410  72.410  68.000  
Torsion^{d}  V_{0}  V_{1}  V_{2}  
CCCC  10.6400  0.0000  10.6400 
Figure 1: Staring structure of two toroid with different radii, the smallest stable toroid and the largest toroid used in our calculation. 
Toroids with small radius are highly strained. To stablize the structure, harmonic bond interactions are used at the early stage of the minimization. The more accurate Morse potential that allows bond breaking are used at the latter stage of minimization. By doing so, we can avoid the bias built in when the starting structure was created. This is important for tracking down the transition radius that separates stable toroids (though highly strained) from the unstable toroids (under Morse bond interaction, the structure flies apart). Figure 2 is the strain energy per atom (relative to infinite long straight (10, 10) tube) versus 1/R^{2}. For toroids with different radius, different final structures resulted. In the plot, we can identify three transition radii, associated with four structural regions.
Figure 2: Strain energy per atom versus curvature. 
For toroids with radii larger than R_{s} = 183.3 (Å) (corresponds to (10, 10, 943) toroid with 18,860 atoms), after molecular dynamics simulation and energy minimization, smooth toriod is the only stable structure. This corresponds to the elastic bending of isolated (10, 10) tube.
For toroids with radii smaller than R_{s} = 183.3 (Å) and larger than 109.6 (Å) ((10, 10, 564) with 11280 atoms), the optimum structures obtained through minimization are smooth toroids without buckles. However, after 20 pico seconds of molecular dynamics equilibration at 300 (K), numerous small dents appeared along the inner wall. Take a snapshot of dynamics trajectory as the starting point of structural minimization, we found an interesting phenomena. During the minimization, small dents diffused along the inner tube and nucleated into larger dents when they meet. This nucleation of deformations continues, until the optimum structure resulted. The optimum structures usually have a number of buckles almost uniformly spaced along the tube.
Figure 3 are the snapshots at the late stage of minimization for (10, 10, 564) (toroid with radius of 109.6 (Å) and 11280 atoms). Looking at the lower left quarter of each ring, we can clearly identify the diffusion of small dents. These small dents eventually moved toward the larger dent as the minimization progresses, and combined with the large dent. The snapshots are numbered according to the minimization sequence. Snapshot with smaller numbering represents structure at earlier stage of minimization. Comparing to the smooth toroids, these structures have lower strain energy per atom. This is due to the stretching of the outer surface and compression of the inner surface. Knee like buckle relaxes compression over large region at the expense of increased local strain.
Figure 3: Snapshots of structural minimization; numbered according to the minimization stages to illustrate nucleation of small buckles into large buckle. 
Figure 4 gives a close look at a buckle, which is cut out from a optimized toroid with one buckle. At the center of the buckle, tube wall collapsed completely. The closest distance between atoms in opposite tube walls is 3.3 (Å), comparable to the distance between adjacent layers of graphite. A short distance away from the collapsed point, the tube stays almost circular. Rooms created for the inner wall at the buckles relax the stretch and compression along the rest of the toroid.
Figure 4: Close look at a buckle. 
There is a strong correlation between the number of buckles and the curvature of the toroids. The higher the strain (curvature) is, the more buckles appeared in the final structure of minimization. However, for each toroid within this region, there are many stable final structures with different number of buckles, each resulted from different starting sructure. This suggests that there exist many meta stable structures for toroids in this region. The fact that the curves towards small radius in Fig. 2 are not smooth suggests that we are not connecting the points with optimum number of buckles. Generally, if we increase the radius (thus reduce strain) we get structures with smaller number of buckles and when we approach the smooth region, we should get only structures with single buckle.
In order to track down the transition point, we created structures with different number of buckles as starting point of minimization. The buckles are uniformly distributed along the circumferences. To create a buckle, we added artificial harmonic constraint on two atoms in the opposite wall of the tube to pull together the inner wall and the outer wall. After the structures are minimized to lower RMS force, where the structures are stable under Morse bond interaction, we remove the constraints and switch harmonic bond potential back to Morse potential to further optimize the structures. We could just heat up the initial structures by using molecular dynamics and then anneal them down to zero temperature. However, given the size of the toroids in the transition, the long time that takes to anneal each structure, and the fact that there could be several stable structures associated with different number of buckles, it is impractical to do so. Figure 5 shows the transition region where smooth toroids and toroids with different number of buckles coexist. Points with same number of buckles are connected into lines. It clearly shows the overlap and shifts of lines with different number of buckles.
Figure 5: Strain energy per atom versus 1/R^{2} at the transition region where smooth toroids and toroids with different number of buckles coexist. 
Towards the transition point beyond which smooth toroids resulted, we are able to create stable structures with four buckles, three buckles, two buckles and one buckle. At region close to the transition radius R_{s}, only the one buckle toroids have the smallest strain energy per atom. Figure 6 shows the buckled structures in this region.
If we further increase the curvature (decrease radius), at R_{k}=109.6 (Å), (corresponds to (10, 10, 564) toroid with 11,280 atoms), only toroids with various number of buckles exist. At even higher curvature, the toroids are flatterned. In this region, there are no smooth toroids, due to the high strain built in the compression of inner wall and tension of the outer wall. Further decrease the radius down to the point of R_{b} = 38.9 (Å) (corresponds to (10, 10, 200) toroid with 4,000 atoms), the structure breaks and atoms fly apart in the course of minimization. Figure 7 shows toroids with more than eight buckles to the smallest toroid that can stand the built in strain.
Figure 7: Collection of buckled toroids. 
Consider the (10, 10) tube as thin elastic rods, then the toroids are rings of thin rods. Assuming k as the Young's modulus of the (10,10) tube, I the moment of inertia about the axis parallel to tube cross section, the strain energy of the rings are given by
Taking r_{out} = 16.70 (Å), the intertube distance of (10, 10) SWNT crystals, r_{in} = 10.5 (Å), which assumes 13.6 (Å) as the radius of (10, 10) tube, we get Young's modulus of the 913 (GPa) for toroids of large radius, which compares to the experimental value of 1280+/ 0.6 (GPa)^{20} for multiwalled carbon nano tubes (MWNT).
We have investigated energetics and structures of (10, 10, n) toroids, three transition radii are found that define the regions with different stable structures. Below R_{b} = 38.9 Å, there is no stable (10,10,l) single wall nano rings, betweeen R_{b} = 38.9 Å and R_{k}= 109.6 Å only the toroids with buckles are stable. With R_{s}=183.3 Å, Between R_{k} and R_{s} both circular and kinked structures are stable. In this region the kinked ones are energetically more favorable. Above R_{s}, the circular structures are energetically favorable, but structures with kinks are also possible. The optimum number of buckles for each structure evolves through the propagation of buckles and their coalescence to new ones on the inner portion of the tori. Based on classical elastic theory analysis, we calculated the modulus of different regions. The calculated Young's modulus along the tube axis of (10,10) tube is found to be 913 GPa from these calculations.
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